A conjugated diene polymer plays a very important industrial role and is produced by polymerizing a conjugated diene compound (monomer). A number of polymerization catalysts used for the polymerization of a conjugated diene compound have been proposed. Particularly, many polymerization catalysts have been studied and developed in order to produce a conjugated diene polymer with a high cis-1,4-bond content having high performance in thermal and mechanical properties. For example, composite catalysts having, as a major component, a transition metal composition containing a transition metal such as nickel, cobalt, and titanium have been known, and some of these are widely used industrially as a polymerization catalyst for butadiene, isoprene, and the like (for example, refer to Non-patent Document 1 and Patent Document 1).
In addition, as a polymerization catalyst exhibiting high polymerization activity which is capable of producing a conjugated diene polymer with a higher cis-1,4-bond content, a composite catalyst comprising a rare earth metal-containing compound and an organometallic compound of a metal in the Group I to III has been researched and developed. For example, Patent Documents 2 to 4 disclose a method of producing a conjugated diene polymer by using a catalyst comprising a rare earth element-containing compound, methyl almoxane (MAO), an organoaluminum compound, and a halogen-containing compound. However, it has been still difficult to produce a polymer with a sufficiently narrow molecular weight distribution and a sufficiently high cis-1,4-bond content by using the catalysts disclosed in these Patent Documents. Moreover, a rubber composition produced from a conjugated diene polymer with a comparatively wide molecular weight distribution and a low cis-1,4-bond content cannot satisfy many characteristics such as mechanical properties, crack-growth resistance, and abrasion resistance, required for a rubber used for tires.    [Patent Document 1] JP-B-37-8193    [Patent Document 2] JP-A-10-306113    [Patent Document 3] JP-A-11-35633    [Patent Document 4] JP-A-2000-34320    [Non-patent Document 1] Ind. Eng. Chem., 48,784 (1956)